it depends how you define the group action $\displaystyle *$. if you define that for all $\displaystyle x,g \in G: \ g*x=gx,$ then $\displaystyle Gx=G.$ but if you define $\displaystyle g*x=gxg^{-1},$ for all $\displaystyle g,x \in G,$ then $\displaystyle Gx=\{gxg^{-1}: \ g \in G \},$
which is the conjugacy class of $\displaystyle x.$
thanks for the help so far.
I think * was multiplication in my case and so Gx=G kinda seems more appropriate.
What I still don't get however is what the actual different between Gx and Gg would be if X=G?
Say X doesn't equal G, would Gx and Gg differ then?
This is where g is in G